XFEL and HHG interaction with matter: Effects of ultrashort pulses and random spikes

The theory of photoionization describing the interaction of x-ray free-electron laser (XFEL) pulses and high-harmonic-generated (HHG) radiation is generalized to ultrashort laser pulses, where the concept of the standard ionization probability per unit time in Fermi’s golden rule and inEinstein’s theory breaks down.Numerical calculations carried out in terms of a generalized photoionization probability for the total duration of pulses in the near-threshold regime demonstrate essentially nonlinear behavior, while absolute values may change by orders of magnitude for typical XFEL and HHG pulses. XFEL self-amplified spontaneous emission pulses are analyzed to reveal general features of photoionization for random and regular spikes: the dependences of the nonlinear photoionization probability on carrier frequency and spike duration are very similar, allowing an analytical expectation value approach that is valid even when there is only limited knowledge of random and regular parameters. Numerical simulations carried out for typical parameters demonstrate excellent agreement. © 2021 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http:// creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0046040 Plasma physics is concerned with the properties of a statistical system containing many charged particles and in which long-range Coulomb forces play an important role in establishing collective phenomena. As the density is increased and the temperature lowered, the plasma begins to exhibit features characteristic of condensed matter, where short-range as well as long-range forces are important. In this regime, the so-called “strongly coupled many-particle Coulomb system,” quantum statistics, dynamic effects, and the interplay between atomic, molecular, and nuclear physics become major issues. Associated with these properties is the emergence of features such as insulator-to-metal transitions, order–disorder transitions, and chemical separation. Themicroscopic properties of these plasmas depend on phase transitions, which in turn affect the macroscopic properties of the plasma through the rates of elementary and transport processes. With the rapid development of powerful laser installations, highenergy-density laser-produced plasmas, strongly coupled plasmas, and warm dense matter (matter near solid density, with temperatures of the order of the Fermi energy) have becomemajor areas of research in high-energy-density physics, relevant to a wide range of topics: dense-plasma astrophysics; the interiors of white and brown dwarfs, the Sun, neutron stars, and giant planets; dense laboratory plasmas (including metals, alloys, and semiconductors); and a variety of plasmas produced by diamond-anvil cells, metal vaporization, shock compression, and pinches.Dense strongly coupled plasmas andwarm dense matter are exotic conditions of matter that are extremely challenging to investigate, and there is a great need for experimental studies to advance the subject. Inmany cases, x-ray emission from the sample under study itself provides a unique characterization: opacity is low, allowing recordings of volumetric self-emission of the sample. However, the matter of interest here (strongly coupled plasmas and warm dense matter) typically has a temperature of the order of the Fermi energy (the order of 10 eV). Target self-emission in the x-ray range is therefore negligible. To circumvent the absence of volumetric x-ray self-emission, essentially two methods have been developed. One of these employs Matter Radiat. Extremes 6, 034001 (2021); doi: 10.1063/5.0046040 6, 034001-1 ©Author(s) 2021 Matter and Radiation at Extremes LETTER scitation.org/journal/mre

Plasma physics is concerned with the properties of a statistical system containing many charged particles and in which long-range Coulomb forces play an important role in establishing collective phenomena. As the density is increased and the temperature lowered, the plasma begins to exhibit features characteristic of condensed matter, where short-range as well as long-range forces are important. In this regime, the so-called "strongly coupled many-particle Coulomb system," quantum statistics, dynamic effects, and the interplay between atomic, molecular, and nuclear physics become major issues. 1,2 Associated with these properties is the emergence of features such as insulator-to-metal transitions, order-disorder transitions, and chemical separation. The microscopic properties of these plasmas depend on phase transitions, which in turn affect the macroscopic properties of the plasma through the rates of elementary and transport processes.
With the rapid development of powerful laser installations, highenergy-density laser-produced plasmas, strongly coupled plasmas, and warm dense matter (matter near solid density, with temperatures of the order of the Fermi energy) have become major areas of research in high-energy-density physics, relevant to a wide range of topics: 3-10 dense-plasma astrophysics; the interiors of white and brown dwarfs, the Sun, neutron stars, and giant planets; dense laboratory plasmas (including metals, alloys, and semiconductors); and a variety of plasmas produced by diamond-anvil cells, metal vaporization, shock compression, and pinches. Dense strongly coupled plasmas and warm dense matter are exotic conditions of matter that are extremely challenging to investigate, and there is a great need for experimental studies to advance the subject. In many cases, x-ray emission from the sample under study itself provides a unique characterization: opacity is low, allowing recordings of volumetric self-emission of the sample. However, the matter of interest here (strongly coupled plasmas and warm dense matter) typically has a temperature of the order of the Fermi energy (the order of 10 eV). Target self-emission in the x-ray range is therefore negligible.
To circumvent the absence of volumetric x-ray self-emission, essentially two methods have been developed. One of these employs high-intensity lasers: the interaction with the near-solid-density matter generates suprathermal electrons that in turn ionize inner atomic shells of the "warm" target material, with the subsequently emitted radiation being in the x-ray spectral range. [11][12][13] For example, for K-shell ionization of partially ionized copper and the corresponding x-ray K α emission, we have where the electronic configurations are described by the number of electrons (upper index) present in given shells (K, L, M), x indicates the degree of ionization of the heated sample (e.g., x 18 for singly ionized copper, x 17 for doubly ionized copper, . . .), e hot is an electron with "suprathermal" energy, and ω Kα is the frequency of the K alpha transition. The other method is similar, but instead of suprathermal electrons, a pulse from an x-ray free-electron laser (XFEL) 14-17 is used: where ω XFEL is the frequency of the XFEL photons and e photo is a photoionized electron. The tuning capabilities of XFELs (in particular with regard to frequency and pulse duration) make them exceptional tools for fundamental studies. 9,10 According to standard quantum mechanical perturbation theory, the total probability W of a transition during an electromagnetic pulse is equal to the probability per unit time w multiplied by the pulse duration τ. The probability W corresponds to Fermi's famous golden rule that connects the total probability with the probability per unit time via the cross-section of the photo-process σ photo and the photon flux density j ph according to w W/τ } σ photo j ph . This relation corresponds to Einstein's theory of stimulated emission and absorption: the number of photon absorptions per unit time N abs is described by the Einstein coefficient of stimulated absorption B abs , i.e., N abs N low B abs j ph , where N low is the number of atoms in the state that is absorbing the photon flux j ph . Correspondingly, the number of stimulated emissions per unit time is N em N up B em j ph , where N up is the number of atoms in the upper state and B em is the Einstein coefficient of stimulated emission. These relations are at the heart of the XFEL interaction with matter and are explicitly used in most laser-matter simulations. [18][19][20][21][22][23][24] However, technological progress in the generation of ultrashort XFEL pulses with durations of the order of a few tens of attoseconds, [25][26][27] the "fine structure" of self-amplified spontaneousemission (SASE) pulses 25,28-31 (spike durations of the order of 0.1-0.2 fs), and high-harmonic generation (HHG) with durations down to 67 as 32 raise questions about the applicability of the basic quantum mechanical framework of a constant probability per unit of time. In fact, the standard probability per unit time w corresponds to pulse durations long enough to be approximated by a single mode of radiation with constant w. An ultrashort pulse, however, is composed of different spectral components, each of which corresponds to a different w value (in an envelope of modes) with typical frequency width of the order of 1/τ. In this case, the total probability is determined by the sum of specific w values, which is not proportional to the duration τ.
To develop a physically transparent description, let us consider ultrashort pulse theory in the framework of Fermi's equivalentphoton concept. 33 An ultrashort pulse can then be represented as a composition of monochromatic modes (equivalent photons) with amplitudes equal to the Fourier pulse transformation. After multiplication of a specific mode by the cross-section of the elementary atomic process and summation over all frequencies in the pulse, we arrive at the following expression for the total probability: 34 where c is the speed of light, ω c is the carrier frequency, E(ω) is the Fourier transform of the electric field strength of the incident pulse, and σ abs is the usual absorption cross-section. According to the optical theorem and in the framework of the dipole approximation, the absorption cross-section spectrum can be expressed through the dynamical polarizability α: 35 In turn, the dynamical polarizability (in atomic units) can be calculated using the real [f 1 (ω)] and imaginary [f 2 (ω)] parts of the atomic scattering factor: 36 and finally the absorption cross-section spectrum can be expressed via the imaginary part of the scattering factor, which is tabulated over a wide range of energies and elements: 37 The importance of the nonlinearity in the photoionization process can best be demonstrated analytically by employing Kramers' formula for the photoionization, and the squared modulus of the Fourier component of a corrected Gaussian pulse shape, 38 where n is the principal quantum number, Z eff is the effective charge of the ion core, E 0 is the electric field amplitude, ω and ω c are the current and carrier frequencies, and τ is the pulse duration. Kramers' formula has a high precision even for the ground state of H-like ions 39 and allows one to consider any quantum state n (whereas the exact Sommerfeld formula is only applicable to the H-like ground state 1s 2 S 1/2 ). Inserting Eqs. (7) and (8) into Eq. (3), we obtain whereτ τI n ,ω ω c /I n , I n Z 2 eff /2n 2 , E n Z 3 eff /2n 5/2 , and erfc is the complementary error function. We can see from Eq. (9) that the ionization probability depends in a highly nonlinear manner on frequency and pulse duration. In the long-pulse limit, ifω < 1, we obtain W n (τ ≫ 1) → 0, whereas in the opposite case whenω > 1, we find W n (τ ≫ 1) }τ.
Handling the random nature of the SASE spikes is difficult for both theory and experiment. For example, the necessary data normalization of absorption measurements for random pulses is quite challenging: either a very large number of pulses have to be accumulated, 40 which is very inconvenient for samples heated with optical lasers, where timing issues are important, or a twofold normalization procedure has to be applied 5 to find a compromise between persistent random features and typically low shot numbers (5-20; see, e.g., Ref. 5) for warm dense matter samples. From a theoretical point of view, random parameters in simulations are challenging, since the differentiation of general or specific features in the XFEL-matter interaction is not a trivial task.
To obtain insight into the general features driven by random pulses, we represent an XFEL pulse with carrier frequency ω c as a series of n modulated Gaussian spikes with envelopes of Gaussian shape: where t j and φ j are the time of maximum and the initial phase of the jth pulse. The characteristic durations of a single spike and envelope are denoted by τ and T, respectively. Here and throughout the following, we consider E 0 to be equal to 1 a.u. If it is assumed that the spikes do not overlap with each other, then the Fourier transform of the pulse train according to Eq. (10) is given by while the time at maximum of the jth regular spike is given by It should be noted that the approximate time duration of the envelope is 8T, where 8 ≈ 6-8. The model indicates that the initial phases of regular spikes are equal to zero and that the corresponding t j parameters are equal to t 0 j . In the case of random spikes, the phases φ j are distributed uniformly over the interval [−π, π], while the parameters t j are uniformly distributed over the intervals [t 0 j − Δ t , t 0 j + Δ t ]. Δ t satisfies the following constraint related to the overlap of spikes: Since we seek to apply the signal model also with random parameters to the description of the ionization process, it is necessary to obtain the expectation value M |E(ω)| 2 of the squared modulus of the Fourier transform: where ε tot is the power spectral density. We have obtained the expectation formula for the squared modulus of the spectral density under the assumption that the probability densities p(t j ) and p(φ j ) are constant and equal to 1/2Δ t and 1/2π, respectively (for more details, see the Appendix).
The spectrum of regular/random spikes and the expectation values are shown in Fig. 1. Here and in the following figures, values are divided by the total signal energy ε tot +∞ −∞ E 2 (t) dt [in order to remove the trivial dependence on total energy related to pulse duration; see the discussion below of Eq. (16)]. The curve of the expectation value has a Gaussian shape. The magnitude of the regular spikes is approximately two times larger because in this case all pulse energy is concentrated into narrow spectral lines.
A random partial-coherence approach was employed in Ref. 41 to calculate a set of pulse shapes and numerically retrieve the characteristics measured at the FLASH FEL. 42 The approach adopted in this article is different in that it identifies common features in any XFEL pulse shape and hence is more generalized.
Let us now study the impact of spikes within a laser pulse on the ionization probability. Figure 2 shows the ionization probability W of solid aluminum for different carrier frequencies. For better demonstration, the probability W is divided by the power spectral density of the pulse ε tot [see Eq. (15)]. The curves demonstrate that for all carrier frequencies (above or below threshold), the dependence of the total probability W(τ)/ε tot (τ) on the spike duration τ is not constant: the nonlinear behavior is qualitatively equivalent to the single-pulse effects discussed in relation to Eqs. (7)-(9) and distinctly different from the standard theory of photoionization. The nonmonotonic behavior is very pronounced for τ ≤ 0.2 fs. This nonlinear behavior manifests itself in W(τ)/ε tot (τ) ≠ const and is of particular practical importance, since typical spike durations in XFEL SASE pulses are of the order of 0.1-0.2 fs. 25,[28][29][30][31] Let us now explore the physical origin of the nonlinear dependences. The standard long-pulse behavior, namely, is obtained from Eq. (3) in the long-pulse limit and for carrier frequencies that are larger than the ionization threshold: this corresponds to Fermi's golden rule of constant probability per unit time. Figure 2 shows that for carrier frequencies lower than the ionization threshold, the dependence of W(τ)/ε tot (τ) is not constant and tends to zero for increasing spike duration (see the curves for ω c 1559 eV and ω c 1555 eV). At some limiting values of τ, the signal band becomes too narrow such that all spectral components in Eq. (11) are lower than the ionization threshold. At carrier frequencies higher than the ionization threshold (see the curves for ω c 1565, 1570, and 1575 eV), the monotonically increasing values of W(τ)/ε tot (τ) for increasing spike duration τ are due to the narrowing of the signal band: the longpulse limit is continuously approached for each single spike, and all spectral components are located in the vicinity of the carrier frequency. At large spike durations, all the signal band is above the ionization threshold, and the W(τ)/ε tot (τ) dependence saturates. This is the regime where Eq. (16) applies.
To demonstrate the principal effects of the short-pulse theory of the inner-shell electron ionization process of warm dense matter outlined above, we develop a simple but finite-temperature absorption model. Taking into account the Fermi distribution for the cross-section spectrum calculation at arbitrary temperature, we obtain with where Θ is the electron temperature in eV. Equations (17) and (18) are applied to the ionization of the K shell of solid aluminum, with the K-shell ionization energy taken as Zω ion 1559.6 eV 43 and the Fermi temperature taken as Θ f 11.7 eV. 44 The coefficients C 1 and C 2 are calculated according to data on the spectral cross-section at room temperature and employing Eq. (6) and the imaginary part of the atomic form factor as given in Ref. 37: C 1 7.4 3 −4 a.u. and C 2 3.8 a.u. Finally, the chemical potential μ(Θ) is evaluated numerically 45 for the electron density in the conduction band of solid aluminum (n e 1.8 3 10 23 cm −3 ). Figure 3 shows the spectral K-shell absorption cross-section of solid aluminum for different temperatures. It can clearly be seen that the absorption edge becomes smoother with increasing temperature. We note that the slope of the absorption cross-section can be employed as a temperature diagnostic in warm dense matter. 6 Let us now apply the simple finite-temperature K-shell absorption model of aluminum to demonstrate the principal effects on the ionization probability induced by x-ray pulses consisting of spikes. Figure 4 displays the calculated results for the ionization probability as a function of the carrier frequency ω c for regular spikes with different single-spike durations. When the carrier frequency is below the ionization threshold, short-pulse spikes effectively change the apparent slope of the spectral absorption probability W(ω c )/ε tot (ω c ): the shorter the pulse duration of the spikes, the flatter is the absorption probability. The pronounced probabilities below threshold for short pulses are due to the broadening of the Fourier transform of the total spectral pulse, which provides spectral components exceeding the ionization threshold.
To evaluate the relative importance of temperature and ultrashort pulses, Fig. 5 shows the spectral absorption probability for a single Gaussian pulse and an x-ray pulse consisting of regular spikes. For example, the relative impact of temperature and short-pulse effects can be obtained by comparing the ionization probability (divided by ϵ tot ) for a single Gaussian pulse with a pulse duration τ G 20 fs at temperature Θ 2.4 eV with that for an x-ray pulse containing spikes with the same envelope duration T 20 fs and typical spike duration τ 0.1 fs but at room temperature Θ 0.025 eV. The effects of temperature and of ultrashort pulses are similar in this example, since the respective curves have a similar appearance.
We now explore the general features of photoionization driven by different types of spikes. Figure 14) and (15)]. It is observed that the absorption cross-section profile at finite temperature becomes smoother with decreasing regular spike duration, and the same tendency also holds for the random spikes in Fig.  6. It is likewise observed that the results of the expectation value approach are in very good agreement with those of the numerical calculations.   Moreover, the simulations demonstrate that the differences between the ionization probabilities for random and regular spikes are rather small (the fact that the absolute values for regular spikes are slightly lower is a particular case and is related to the simultaneous features of equidistant spikes and zero phases). Therefore, random simulations and the expectation value approach are useful approximations for any cases of interest, i.e., cases that are neither totally regular nor totally random. This is of great practical importance, since there is usually only limited knowledge of random and regular parameters in experiments.
In summary, we have generalized photoionization theory to ultrashort x-ray pulses, where the standard approach of constant probability per unit time (Fermi's golden rule together with Einstein's theory of stimulated emission and absorption) does not apply. A model has been developed where an XFEL pulse is represented by an envelope with a carrier frequency and a series of spikes for two different regimes: (i) spikes with random initial phases and random times for spike maxima and (ii) regular spikes. An analytical model of finite-temperature dense matter using the Fermi distribution has allowed us to explore the effects of ultrashort pulses on dense matter under extreme conditions. In the vicinity of the ionization threshold, the dependence of the ionization probability on carrier frequency rises from zero to a constant value for solid matter at room temperature. By contrast, calculations for a single Gaussian pulse with a duration equal to the XFEL pulse envelope recover the standard dependence in the form of a sharp step function. Simulations for typical target and laser parameters have revealed general properties for x-ray pulses containing spikes (e.g., XFEL SASE pulses): the dependences of the nonlinear photoionization probability on carrier frequency and spike duration for random spikes and regular spikes are very similar, indicating that the model developed here can be applied to a wide variety of practical cases even if random and regular contributions are not well known (as is typical in experiments). As almost all x-ray pulse-matter interactions proceed from photoabsorption, the generalized ultrashort pulse model and the analytical expectation value approach are of interest for a wide variety of simulations of the interaction of XFEL beams with matter.